Integrand size = 27, antiderivative size = 378 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\frac {2 \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 (d-a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{3 \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}} \]
2/3*(c*(-a*e+d)*cos(x)-b*(-a*e+d)*sin(x))/(a^2-b^2-c^2)/(a+b*cos(x)+c*sin( x))^(3/2)+2/3*(c*(4*a*d-a^2*e-3*(b^2+c^2)*e)*cos(x)-b*(4*a*d-a^2*e-3*(b^2+ c^2)*e)*sin(x))/(a^2-b^2-c^2)^2/(a+b*cos(x)+c*sin(x))^(1/2)+2/3*(4*a*d-a^2 *e-3*(b^2+c^2)*e)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arcta n(b,c))*EllipticE(sin(1/2*x-1/2*arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+( b^2+c^2)^(1/2)))^(1/2))*(a+b*cos(x)+c*sin(x))^(1/2)/(a^2-b^2-c^2)^2/((a+b* cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/3*(-a*e+d)*(cos(1/2*x-1/2*ar ctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arctan(b,c))*EllipticF(sin(1/2*x-1/2*arc tan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b*cos(x )+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a^2-b^2-c^2)/(a+b*cos(x)+c*sin(x)) ^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.73 (sec) , antiderivative size = 5554, normalized size of antiderivative = 14.69 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\text {Result too large to show} \]
Time = 1.74 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3635, 27, 3042, 3635, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {b e \cos (x)+c e \sin (x)+d}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {b e \cos (x)+c e \sin (x)+d}{(a+b \cos (x)+c \sin (x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}-\frac {2 \int -\frac {3 \left (a d-\left (b^2+c^2\right ) e\right )-b (d-a e) \cos (x)-c (d-a e) \sin (x)}{2 (a+b \cos (x)+c \sin (x))^{3/2}}dx}{3 \left (a^2-b^2-c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 \left (a d-\left (b^2+c^2\right ) e\right )-b (d-a e) \cos (x)-c (d-a e) \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}}dx}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 \left (a d-\left (b^2+c^2\right ) e\right )-b (d-a e) \cos (x)-c (d-a e) \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}}dx}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2 \int -\frac {3 d a^2-4 \left (b^2+c^2\right ) e a+\left (b^2+c^2\right ) d+b \left (-e a^2+4 d a-3 \left (b^2+c^2\right ) e\right ) \cos (x)+c \left (-e a^2+4 d a-3 \left (b^2+c^2\right ) e\right ) \sin (x)}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 d a^2-4 \left (b^2+c^2\right ) e a+\left (b^2+c^2\right ) d+b \left (-e a^2+4 d a-3 \left (b^2+c^2\right ) e\right ) \cos (x)+c \left (-e a^2+4 d a-3 \left (b^2+c^2\right ) e\right ) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 d a^2-4 \left (b^2+c^2\right ) e a+\left (b^2+c^2\right ) d+b \left (-e a^2+4 d a-3 \left (b^2+c^2\right ) e\right ) \cos (x)+c \left (-e a^2+4 d a-3 \left (b^2+c^2\right ) e\right ) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {\frac {\left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-\left (a^2-b^2-c^2\right ) (d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-\left (a^2-b^2-c^2\right ) (d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {\frac {\frac {\left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) (d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) (d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) (d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {\left (a^2-b^2-c^2\right ) (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {\left (a^2-b^2-c^2\right ) (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}\) |
(2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)*(a + b* Cos[x] + c*Sin[x])^(3/2)) + ((2*(c*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Cos[x ] - b*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Sin[x]))/((a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]]) + ((2*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*EllipticE [(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2] )] - (2*(a^2 - b^2 - c^2)*(d - a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqr t[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + S qrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]])/(a^2 - b^2 - c^2))/(3*(a^ 2 - b^2 - c^2))
3.6.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])] Int[Sqrt[a/(a + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] Int[1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 , 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] , x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] , x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ A*b - a*B, 0]
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) *Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(4019\) vs. \(2(406)=812\).
Time = 142.43 (sec) , antiderivative size = 4020, normalized size of antiderivative = 10.63
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(4020\) |
| parts | \(\text {Expression too large to display}\) | \(1032654\) |
-(-(-b^2*sin(x-arctan(-b,c))-c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))*co s(x-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)/(b^2+c^2)^(1/2)*(b^4*sin(x-arct an(-b,c))^4+2*b^2*c^2*sin(x-arctan(-b,c))^4+c^4*sin(x-arctan(-b,c))^4-2*a^ 2*b^2*sin(x-arctan(-b,c))^2-2*a^2*c^2*sin(x-arctan(-b,c))^2+a^4)*(cos(x-ar ctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*(si n(x-arctan(-b,c))^2*b^2*e+sin(x-arctan(-b,c))^2*c^2*e+e*a*sin(x-arctan(-b, c))*(b^2+c^2)^(1/2)+d*sin(x-arctan(-b,c))*(b^2+c^2)^(1/2)+a*d)/(2*(((b^2+c ^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*sin(x-arctan (-b,c))^2*a*b^2*e+2*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan( -b,c))^2)^(1/2)*sin(x-arctan(-b,c))^2*a*c^2*e-(cos(x-arctan(-b,c))^2*((b^2 +c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*sin(x-arctan(-b,c))^3* b^2*e-(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+ c^2))^(1/2)*sin(x-arctan(-b,c))^3*c^2*e-(((b^2+c^2)^(1/2)*sin(x-arctan(-b, c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*sin(x-arctan(-b,c))^2*b^2*d-(((b^2+c^2 )^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*sin(x-arctan(- b,c))^2*c^2*d-e*a^2*sin(x-arctan(-b,c))*(cos(x-arctan(-b,c))^2*((b^2+c^2)^ (1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)-d*a^2*(((b^2+c^2)^(1/2)*sin( x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+2*d*a*sin(x-arctan(-b,c))* (cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^ (1/2))/(b^4*sin(x-arctan(-b,c))^3+2*b^2*c^2*sin(x-arctan(-b,c))^3+c^4*s...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.37 (sec) , antiderivative size = 4057, normalized size of antiderivative = 10.73 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\text {Too large to display} \]
1/9*((sqrt(2)*(I*(a^2*b^3 + 3*b^5 - a^2*b*c^2 - 3*b*c^4)*d - (a^2*c^3 + 3* c^5 - (a^2*b^2 + 3*b^4)*c)*d + 2*I*(a^3*b^3 - 3*a*b^5 - a^3*b*c^2 + 3*a*b* c^4)*e - 2*(a^3*c^3 - 3*a*c^5 - (a^3*b^2 - 3*a*b^4)*c)*e)*cos(x)^2 - 2*sqr t(2)*(-I*(a^3*b^2 + 3*a*b^4 + 3*a*b^2*c^2)*d - (3*a*b*c^3 + (a^3*b + 3*a*b ^3)*c)*d - 2*I*(a^4*b^2 - 3*a^2*b^4 - 3*a^2*b^2*c^2)*e + 2*(3*a^2*b*c^3 - (a^4*b - 3*a^2*b^3)*c)*e)*cos(x) - 2*(sqrt(2)*(-I*(3*b^2*c^3 + (a^2*b^2 + 3*b^4)*c)*d - (3*b*c^4 + (a^2*b + 3*b^3)*c^2)*d + 2*I*(3*a*b^2*c^3 - (a^3* b^2 - 3*a*b^4)*c)*e + 2*(3*a*b*c^4 - (a^3*b - 3*a*b^3)*c^2)*e)*cos(x) + sq rt(2)*(-I*(3*a*b*c^3 + (a^3*b + 3*a*b^3)*c)*d - (3*a*c^4 + (a^3 + 3*a*b^2) *c^2)*d + 2*I*(3*a^2*b*c^3 - (a^4*b - 3*a^2*b^3)*c)*e + 2*(3*a^2*c^4 - (a^ 4 - 3*a^2*b^2)*c^2)*e))*sin(x) + sqrt(2)*(I*(a^4*b + 3*a^2*b^3 + 3*b*c^4 + (4*a^2*b + 3*b^3)*c^2)*d + (3*c^5 + (4*a^2 + 3*b^2)*c^3 + (a^4 + 3*a^2*b^ 2)*c)*d + 2*I*(a^5*b - 3*a^3*b^3 - 3*a*b*c^4 - (2*a^3*b + 3*a*b^3)*c^2)*e - 2*(3*a*c^5 + (2*a^3 + 3*a*b^2)*c^3 - (a^5 - 3*a^3*b^2)*c)*e))*sqrt(b + I *c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3 *c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3* b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c ^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(I*b^2 + I*c^2) *sin(x))/(b^2 + c^2)) + (sqrt(2)*(-I*(a^2*b^3 + 3*b^5 - a^2*b*c^2 - 3*b...
Timed out. \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\int { \frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{{\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\int { \frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{{\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\int \frac {d+b\,e\,\cos \left (x\right )+c\,e\,\sin \left (x\right )}{{\left (a+b\,\cos \left (x\right )+c\,\sin \left (x\right )\right )}^{5/2}} \,d x \]